Finite Group Databasebeta

← #360665

Subgroups — #360665

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2C2×#1845C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5#14686C211yesmaximal, characteristic
6#11096C214yesmaximal
7#10907C214yesmaximal, complemented
8#10841C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10C22×#117—21no—
11C23×C42C2211yescharacteristic
12C22×#107C2214yes—
13C22×C22⋊C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2285—24no—
17#2285—24no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382—21no—
23C2×#380—21no—
24C2×#380—21no—
25C2×#380—21no—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30#1845—24no—
31#1834—24no—
32C4×C22⋊C8C418yescomplemented
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#117—28no—
46C2×#117—44no—
47C2×#117—44no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42C2×C414yes—
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42Q814yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#107—28no—
74C2×#107—28no—
75C2×#107—44no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8C2×C418yes—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8C2×C414yes—
84C2×C4×C8C2×C414yes—
85#384—22no—
86#384—22no—
87#384—22no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384D414yes—
92#384D414yes—
93#382—28no—
94#380—28no—
95#380—24no—
96#380—24no—
97#380—44no—
98#380—44no—
99#380—44no—
100C4×C4⋊C4—28no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108#340—28no—
109#340—28no—
110#340—44no—
111#336C2×C418yes—
112C25C2×D411yescharacteristic
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C4⋊C414yes—
136C23×C4C4⋊C414yes—
137C23×C4D814yes—
138C23×C4Q1614yes—
139C23×C4C22×C412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×Q812yes—
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C414yes—
158C22×C8C22×C414yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161#117—432no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—28no—
217C2×C42—28no—
218C2×C42—28no—
219C2×C42—28no—
220C2×C42—28no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42C4⋊C414yes—
224C2×C42C4⋊C414yes—
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×Q812yes—
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233#107—416no—
234C22⋊C8—216no—
235C4×C8—28no—
236C4×C8—28no—
237#95—28no—
238#95—28no—
239#95—44no—
240#95—44no—
241#95—44no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24C2×C4⋊C412yes—
255C24C2×D812yes—
256C24C2×Q1612yes—
257C24C22×D411yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—28no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4#107116yescomplemented
332C22×C4C2×C4⋊C412yes—
333C22×C4C2×C4⋊C412yes—
334C22×C4C2×C4⋊C412yes—
335C22×C4C2×C4⋊C412yes—
336C22×C4C2×C4⋊C412yes—
337C22×C4#12012yes—
338C22×C4#12012yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4C2×D812yes—
342C22×C4C2×D812yes—
343C22×C4C2×Q1612yes—
344C22×C4C2×Q1612yes—
345C22×C4#13512yes—
346C22×C4#13512yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4AGL(1,ℤ/8)12yes—
349C22×C4#13712yes—
350C22×C4#13712yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×Q811yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8#11718yescomplemented
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C22⋊C4—416no—
392C22⋊C4—88no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—28no—
398C42—28no—
399C42—416no—
400C42C4×D414yes—
401C42C4×D414yes—
402C42C22⋊Q814yes—
403C42C22⋊Q814yes—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23C2×#10714yes—
439C23C22×C4⋊C411yescharacteristic
440C23C2×#12011yescharacteristic
441C23C2×#12311yescharacteristic
442C23#53211yescharacteristic
443C23#53211yescharacteristic
444C23#53511yescharacteristic
445C23#53611yescharacteristic
446C23C22×D811yescharacteristic
447C23C22×Q1611yescharacteristic
448C23C2×#13511yescharacteristic
449C23C2×#13511yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23C2×AGL(1,ℤ/8)11yescharacteristic
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—232no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—416no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38514yes—
509C2×C4#38514yes—
510C2×C4C2×#10714yes—
511C2×C4C2×#10714yes—
512C2×C4#42714yes—
513C2×C4#42814yes—
514C2×C4#42814yes—
515C2×C4#44714yes—
516C2×C4#44914yes—
517C2×C4#45114yes—
518C2×C4#45214yes—
519C2×C4#48014yes—
520C2×C4#48214yes—
521C2×C4#48414yes—
522C2×C4#48514yes—
523C2×C4C2×#11712yes—
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4#52012yes—
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4#53312yes—
530C2×C4#54412yes—
531C2×C4#54612yes—
532C2×C4#54712yes—
533C2×C4#54812yes—
534C2×C4#55012yes—
535C2×C4#55212yes—
536C2×C4#55312yes—
537C2×C4#55512yes—
538C2×C4#55612yes—
539C8—28no—
540C8—44no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22#228511yescharacteristic
557C22C2×#38511yescharacteristic
558C22#231711yescharacteristic
559C22#243111yescharacteristic
560C22#243211yescharacteristic
561C22#249911yescharacteristic
562C22#250011yescharacteristic
563C22C22×#10711yescharacteristic
564C22C2×#42711yescharacteristic
565C22C2×#42811yescharacteristic
566C22C2×#42811yescharacteristic
567C22C2×#44711yescharacteristic
568C22C2×#45111yescharacteristic
569C22C2×#44911yescharacteristic
570C22C2×#45211yescharacteristic
571C22C2×#48011yescharacteristic
572C22C2×#48211yescharacteristic
573C22C2×#48511yescharacteristic
574C22C2×#48411yescharacteristic
575C22#316911yescharacteristic
576C22#316911yescharacteristic
577C22#317011yescharacteristic
578C22#317011yescharacteristic
579C22#318211yescharacteristic
580C22#318211yescharacteristic
581C22#318411yescharacteristic
582C22#318511yescharacteristic
583C22#320511yescharacteristic
584C22#320711yescharacteristic
585C22#320911yescharacteristic
586C22#321011yescharacteristic
587C22#322611yescharacteristic
588C22#322811yescharacteristic
589C22#323011yescharacteristic
590C22#323111yescharacteristic
591C4—24no—
592C4—42no—
593C4—42no—
594C4—84no—
595C4—84no—
596C4—162no—
597C4—162no—
598C4#185218yes—
599C4#185418yes—
600C4#235212yes—
601C4#235412yes—
602C4#243912yes—
603C4#244112yes—
604C2—44no—
605C2#1387911yescharacteristic
606C2#2081711yescharacteristic
607C2#2082411yescharacteristic
608C2C2×#185211yescharacteristic
609C2C2×#185411yescharacteristic
610C2#2198211yescharacteristic
611C2#2198411yescharacteristic
612C2#2559911yescharacteristic
613C2#2560011yescharacteristic
614C2#2567311yescharacteristic
615C2#2567411yescharacteristic
616C2#2787711yescharacteristic
617C2#2787811yescharacteristic
618C2#2789111yescharacteristic
619C2#2789211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1845C2semidirect
3. C2×#1834C2semidirect
7. #10907C2semidirect
8. #10841C2semidirect
9. #10841C2semidirect
32. C4×C22⋊C8C4semidirect
331. C22×C4#107semidirect
374. C2×C8#117semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22852
C2×#18452
C2×#18342
C2×C4×C22⋊C82
#146862
#110962
#109072
#108412